Sometimes it seems that the difference between their unit fractions is the LCD of two fractions. one-twelfth equals one-eighteenth plus one thirty-sixth
For example, the LCD between twelfths and eighteenths is the difference between their unit fractions. Using circle fractions: 1/12 = 30 degrees, 1/18= 20 degrees. Their difference is 30-20=10 degrees; which is 1/36ths of a circle.
eighths and fourteenths don't seem to work that well
On the other hand sometimes this doesn't work at all - even though it seems kind of close such as a multiple of the LCD or something.
Ideas? Why would this work? When and under which circumstances? Thank you in advance, I will now make sandwiches and wait by my computer.
Origami 1729 Befriending numbers one at a time
Assume $a<b$, and note that:
$$\frac1a-\frac1b=\frac{b-a}{ab} = \frac{b-a}{\gcd(a,b)\cdot\operatorname{lcm}\left(a,b\right)}$$
To see this, you need to know that the product of two numbers equals the product of their gcd and their lcm, which is a standard result in elementary number theory.
Anyway, if you want this result to equal: $$\frac1{\operatorname{lcm}\left(a,b\right)},$$
then you need the fraction to reduce just right, i.e., you need: $$b-a=\gcd(a,b).$$
With your examples, $18-12=6$, and $6$ is the gcd of $18$ and $12$. On the other hand, $14-8=6$, but the gcd of $14$ and $8$ is only $2$. That's why you're left with $3$ in the numerator of: $$\frac18-\frac1{14}=\frac3{56}.$$